Just to be clear; is the issue that you definitely have no SSH access, or just that you don't know how to / feel confident in using it?
As I say, I recently helped out someone on what seemed to be a fairly basic Hostgator shared hosting account, and their cPanel had the ability to configure SSH access. Another actually just had a web based terminal that logged you right in.
If you just don't have that facility, then I guess considering a move to a different host entirely might be worth considering too.
It might also be worth asking your host too. Frankly if they refused to do something as simple as run a simple script for you, then, again, they're not exactly the best.